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Let's say an object weighs 10 N at rest. The normal force here is 10 N as well since that is by how much the object is pushing down on the surface.

When someone tries to pull it upward with a 6 N force, it "relieves" some of the normal force. Since the object is now only pushing down on the surface with 4 N, the normal force is 4 N.

So the proper set-up for an equation here is F(normal) = F(gravity) - F(upward)

But in an elevator, let's say the elevator is accelerating upward at 2 m/s^2. If a person were inside the elevator, why doesn't this "relieve" some of their normal force as well? Instead, I have been told that the proper equation for this situation is F(upward) = F(normal) - F(gravity). Rearranging that equation, F(normal) = F(upward) + F(gravity). So now the normal force is actually much greater as a result of an upward force.

Why is this so?